Copulas: a Review and Recent Developments (2007)

De¯nition (conditional pseudo-copula, Fermanian and Scaillet (2005)). Theconditional pseudo-copula with respect to the sub-algebras A 1 ;::: ;A n ; A; B and associatedwith X is a random function C pseudo (:jA; B) :[0; 1] n ! [0; 1] such thatH(xjB) =C pseudo (F X1 (x 1 jA 1 );::: ;F Xn (x n jA n )jA; B)almost surely for every x =(x 1 ;::: ;x n ) 2 (¡1; 1) n . Such a function is uniqueon the product of values taken by the conditional marginal cumulative distributionfunctions F Xi (:jA i );i=1;::: ;n:The function C pseudo (:jA; B) is called a pseudo-copula because it satis¯es all propertiesof the usual copula, except the condition 3 in the Formal copula de¯nition.Example (conditional pseudo-copulas are not copulas in general, Fermanianand Wegkamp (2004)). Consider the bivariate processfX n = aX n¡1 + ² n ; Y n = bX n¡1 + cY n¡1 + º n g; n =1; 2;:::;where the sequence of innovation terms ² n and º n are independent Gaussian whitenoises. Set A n;1 = ¾(X n¡1 = x n¡1 ), A n;2 = ¾(Y n¡1 = y n¡1 )andB n = ¾((X; Y ) n¡1 =(x; y) n¡1 ). ThenP (Y n